The expression `sqrt(12+6sqrt(3))+sqrt(12-6sqrt(3))` simplifies to

Options

A. 4
B. `2sqrt(3)`
C. `3sqrt(3)`
D. 6

Answer

To simplify the expression \( \sqrt{12 + 6\sqrt{3}} + \sqrt{12 - 6\sqrt{3}} \), we can follow these steps:

### Step 1: Rewrite the terms under the square roots
We start by rewriting \( 12 \) as \( 9 + 3 \) in both square roots:
\[ \sqrt{12 + 6\sqrt{3}} = \sqrt{9 + 3 + 6\sqrt{3}} = \sqrt{(3 + \sqrt{3})^2} \]
\[ \sqrt{12 - 6\sqrt{3}} = \sqrt{9 + 3 - 6\sqrt{3}} = \sqrt{(3 - \sqrt{3})^2} \]

### Step 2: Apply the square root
Now we can take the square roots:
\[ \sqrt{(3 + \sqrt{3})^2} = 3 + \sqrt{3} \]
\[ \sqrt{(3 - \sqrt{3})^2} = 3 - \sqrt{3} \]

### Step 3: Combine the results
Now we combine the results from the two square roots:
\[ \sqrt{12 + 6\sqrt{3}} + \sqrt{12 - 6\sqrt{3}} = (3 + \sqrt{3}) + (3 - \sqrt{3}) \]

### Step 4: Simplify the expression
When we combine the two expressions, the \( \sqrt{3} \) terms cancel out:
\[ (3 + \sqrt{3}) + (3 - \sqrt{3}) = 3 + 3 = 6 \]

### Final Result
Thus, the simplified expression is:
\[ \sqrt{12 + 6\sqrt{3}} + \sqrt{12 - 6\sqrt{3}} = 6 \]

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